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# Mathematical properties of the Platonic solids

One can specify the following mathematical characteristics in each of the five Platonic solids:

1. The radius of the sphere circumscribing the polyhedron;

2. The radius of the sphere inscribed in the polyhedron;

3. The surface area of the polyhedron;

4. The volume of the polyhedron.

## Tetrahedron:  All four faces are equilateral triangles

The radius of a circumscribed sphere of tetrahedron is

,where "a" is side length

The radius of an inscribed sphere of tetrahedron is

The surface area of tetrahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of tetrahedron is

The height of the tetrahedron is determined by the following formula:

The distance to the center of the base of the tetrahedron is determined by the formula:

## Octahedron: All eight faces are equilateral triangles

The radius of a circumscribed sphere
of an octahedron is

,where "a" is side length

The radius of an inscribed sphere of octahedron is

The surface area of octahedron is

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of octahedron is

## Hexahedron (cube): All six faces are squares

The radius of a circumscribed sphere
of cube

,where "a" is side length

The radius of an inscribed sphere
of cube is

The surface area of cube is

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of cube is

## Dodecahedron: All 12 faces are regular pentagon

The radius of a circumscribed sphere
of dodecahedron

,where "a" is side length

The radius of an inscribed sphere
of dodecahedron is

The surface area of dodecahedron is

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of dodecahedron is

## Icosahedron: All 20 faces are equilateral triangles

The radius of a circumscribed sphere of icosahedron

,where "a" is side length

The radius of an inscribed sphere of icosahedron is

The surface area of icosahedron is

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

the volume of icosahedron is

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